{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1996:BQ5OH4J4NREVLMIX2RSABAQDYE","short_pith_number":"pith:BQ5OH4J4","canonical_record":{"source":{"id":"math/9606209","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1996-06-05T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"7541dc3f66f30e2dd33eede108d649dd866bec6e09ac9966bab0f9e0cf66e5a8","abstract_canon_sha256":"3840d5c6f0975b941c54c4939d81f5ba6ede5f501cba22be27ca6618557e14d7"},"schema_version":"1.0"},"canonical_sha256":"0c3ae3f13c6c4955b117d464008203c1024217d4d1b659ab2b54e1595eeb9929","source":{"kind":"arxiv","id":"math/9606209","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9606209","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"arxiv_version","alias_value":"math/9606209v1","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9606209","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"pith_short_12","alias_value":"BQ5OH4J4NREV","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"BQ5OH4J4NREVLMIX","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"BQ5OH4J4","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1996:BQ5OH4J4NREVLMIX2RSABAQDYE","target":"record","payload":{"canonical_record":{"source":{"id":"math/9606209","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1996-06-05T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"7541dc3f66f30e2dd33eede108d649dd866bec6e09ac9966bab0f9e0cf66e5a8","abstract_canon_sha256":"3840d5c6f0975b941c54c4939d81f5ba6ede5f501cba22be27ca6618557e14d7"},"schema_version":"1.0"},"canonical_sha256":"0c3ae3f13c6c4955b117d464008203c1024217d4d1b659ab2b54e1595eeb9929","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:47.337121Z","signature_b64":"uGa20m+Z1foqekhRnb0bHjupSWT304s5Hb7OqaOn5M5jD2NGZx0WuaM8izFFzAqq+Uv4JWoJA9JqGHAdbqgICw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c3ae3f13c6c4955b117d464008203c1024217d4d1b659ab2b54e1595eeb9929","last_reissued_at":"2026-05-18T01:05:47.336501Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:47.336501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9606209","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2r7egV/823Th/1hwmoETREUzMnx1jBHgrkAXBRDjMoI/OAia0/uLVkcrF+RnBfUZUq43VycWEhy5gEDzDv+kCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:57:58.508072Z"},"content_sha256":"00db1265802d9bdfc4a8d81de260b8e4c16c7d26acae9841ca1af9a75b268583","schema_version":"1.0","event_id":"sha256:00db1265802d9bdfc4a8d81de260b8e4c16c7d26acae9841ca1af9a75b268583"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1996:BQ5OH4J4NREVLMIX2RSABAQDYE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Concerning the Bourgain $ell_1$ index of a Banach space","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Edward Odell, Robert Judd","submitted_at":"1996-06-05T00:00:00Z","abstract_excerpt":"A well known argument of James yields that if a Banach space $X$ contains $\\ell_1^n$'s uniformly then $X$ contains $\\ell_1^n$'s almost isometrically. In the first half of the paper we extend this idea to the ordinal $\\ell_1$-indices of Bourgain. In the second half we use our results to calculate the $\\ell_1$-index of certain Banach spaces. Furthermore we show that the $\\ell_1$-index of a separable Banach space not containing $\\ell_1$ must be of the form $\\omega^{\\alpha}$ for some countable ordinal $\\alpha$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9606209","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qBKY1b0RL/9CT7jbfGwlVqKVK3laYKhPlol3mHiz27K++/8ouEQJspO1K+aGjW+n1WcjIZxpBUWIjh+VvmDYCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:57:58.508428Z"},"content_sha256":"dedba1d9bf42b1a10cc98c8bf2e370bcda36eaf0892afbcf4ea215e14f800765","schema_version":"1.0","event_id":"sha256:dedba1d9bf42b1a10cc98c8bf2e370bcda36eaf0892afbcf4ea215e14f800765"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BQ5OH4J4NREVLMIX2RSABAQDYE/bundle.json","state_url":"https://pith.science/pith/BQ5OH4J4NREVLMIX2RSABAQDYE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BQ5OH4J4NREVLMIX2RSABAQDYE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T06:57:58Z","links":{"resolver":"https://pith.science/pith/BQ5OH4J4NREVLMIX2RSABAQDYE","bundle":"https://pith.science/pith/BQ5OH4J4NREVLMIX2RSABAQDYE/bundle.json","state":"https://pith.science/pith/BQ5OH4J4NREVLMIX2RSABAQDYE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BQ5OH4J4NREVLMIX2RSABAQDYE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:BQ5OH4J4NREVLMIX2RSABAQDYE","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3840d5c6f0975b941c54c4939d81f5ba6ede5f501cba22be27ca6618557e14d7","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1996-06-05T00:00:00Z","title_canon_sha256":"7541dc3f66f30e2dd33eede108d649dd866bec6e09ac9966bab0f9e0cf66e5a8"},"schema_version":"1.0","source":{"id":"math/9606209","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9606209","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"arxiv_version","alias_value":"math/9606209v1","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9606209","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"pith_short_12","alias_value":"BQ5OH4J4NREV","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"BQ5OH4J4NREVLMIX","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"BQ5OH4J4","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:dedba1d9bf42b1a10cc98c8bf2e370bcda36eaf0892afbcf4ea215e14f800765","target":"graph","created_at":"2026-05-18T01:05:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A well known argument of James yields that if a Banach space $X$ contains $\\ell_1^n$'s uniformly then $X$ contains $\\ell_1^n$'s almost isometrically. In the first half of the paper we extend this idea to the ordinal $\\ell_1$-indices of Bourgain. In the second half we use our results to calculate the $\\ell_1$-index of certain Banach spaces. Furthermore we show that the $\\ell_1$-index of a separable Banach space not containing $\\ell_1$ must be of the form $\\omega^{\\alpha}$ for some countable ordinal $\\alpha$.","authors_text":"Edward Odell, Robert Judd","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"1996-06-05T00:00:00Z","title":"Concerning the Bourgain $ell_1$ index of a Banach space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9606209","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:00db1265802d9bdfc4a8d81de260b8e4c16c7d26acae9841ca1af9a75b268583","target":"record","created_at":"2026-05-18T01:05:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3840d5c6f0975b941c54c4939d81f5ba6ede5f501cba22be27ca6618557e14d7","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1996-06-05T00:00:00Z","title_canon_sha256":"7541dc3f66f30e2dd33eede108d649dd866bec6e09ac9966bab0f9e0cf66e5a8"},"schema_version":"1.0","source":{"id":"math/9606209","kind":"arxiv","version":1}},"canonical_sha256":"0c3ae3f13c6c4955b117d464008203c1024217d4d1b659ab2b54e1595eeb9929","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c3ae3f13c6c4955b117d464008203c1024217d4d1b659ab2b54e1595eeb9929","first_computed_at":"2026-05-18T01:05:47.336501Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:47.336501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uGa20m+Z1foqekhRnb0bHjupSWT304s5Hb7OqaOn5M5jD2NGZx0WuaM8izFFzAqq+Uv4JWoJA9JqGHAdbqgICw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:47.337121Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9606209","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00db1265802d9bdfc4a8d81de260b8e4c16c7d26acae9841ca1af9a75b268583","sha256:dedba1d9bf42b1a10cc98c8bf2e370bcda36eaf0892afbcf4ea215e14f800765"],"state_sha256":"f600aecc3535792428510b63d72146ba5bbf6fe02ef2e90d9877567078c8db4d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nzUNvfYUwbxZN9BMsdh/BMaibhBW/C9YIU66KosyB7ygFlcv8XT9t9eg+gP11i1aelbt5S2XNonLToFWI2OJAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T06:57:58.510348Z","bundle_sha256":"a61b34c7a7785a58cc601ab3490cdd06b4ca40e6a9f476c0cc193efcc04f3b07"}}